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ON DYNAMICS AND CONTROL
OF MULTI-LINE; FLEXIBLE SPACE MANIPULATORS
W. Gawronski, C.-H.C. Ih, and S.J. Wang
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Jet Propulsion Laboratory, California Institute of Technology
Pasadena, CA 91109
Abstract. In this paper dynamics, inverse
dynamics, and control problems for multi-link
flexible space
manipulators are presented. In
deriving the flexible manipulator dynamics the
following are assumed: flexible deformations are
relatively small; angular rates of the links are
much smaller than their fundamental frequencies;
nonlinear terms (centrifugal and Coriolis forces)
in the flexible manipulator model are the same as
those in the ri id body model. These assumptions
are reasonable or large space manipulators, such
as the space crane. Flexible displacements are
measured
with
respect
to
the
rigid
body
configuration, for which a linear time-varying
system is obtained. The inverse dynamics problem
consists of determination of joint torques, given
tip trajectory, such that joint angles in flexible
configuration are equal to the angles in the rigid
body configuration. The manipulator control system
consists of the feedforward compensation and
feedback control loops. Simulation results of a
two-link space crane with large payload show that
the performance of this linearized dynamics and
control approach is reasonable and robust subject
to parameter variations during slew operations.
the rigid body configuration. Our simulation
results, which include Coriolis and centrifugal
forces, show that the solution of the first inverse
dynamics problem is unstable and sensitive to model
parameter variations. On the other hand, the
solution of the second problem is stable, and
robust to model errors.
The flexible manipulator position control
presented in this paper is motivated by the rigid
robot control approaches, see Ref. 16. Two types of
control loops are implemented, the feedforward
control loop and the feedback control loop.
Feedforward control torques are precomputed from
the flexible inverse dynamics model and applied to
the joints. The manipulator tip is expected to
closely follow the prescribed trajectory. The
error, due to modeling inaccuracy and external
disturbances, is compensated by the feedback loops
with collocated sensors and actuators. The feedback
loops consist of joint control loops and member
stiffening loops. A pole mobility factor is
introduced in the paper and used for determination
of feedback gain.
!
Simulation results show good performance in
terms of tip accuracy and robustness of the
manipulator system.
1. Introduction
The study of dynamics of flexible manipulators
has been reputed in several papers, see Refs.1-11.
In this paper a linear time-varying model of a
manipulator is derived. The flexible displacements
are determined with respect to the rigid body
configuration. Assuming constant parameter values
within an appropriately small time interval, a time
invariant model is obtained. The parameter values
are updated for each time interval.
2. Flexible manipulator dynamics
A finite element approach is used to derive
the flexible body dynamics. Let qi be the
displacement relative to a time-varyin coordinate
frame attached to the rigid body coniguration of
the manipulator, see Figla.
Given the tip
a
trajectory
(and
additional constraints
for
redundant manipulator) the rigid body configuration
is unique for every time instant, hence the
flexible deformation is uniquely defined. Tbe
following assumptions are introduced:
1. Amplitudes of flexible deformations are small
relative to the link length.
2. Articulation rates of manipulator links are much
smaller than the links fundamental frequencies.
3. Rigid body inertia, centrifugal and Coriolis
forces and torques in the flexible manipulator
model are determined from the corresponding rigid
body model.
The inverse dynamics problem for a flexible
manipulator is the problem of the determination of
joint torques such that the end-effector of the
manipulator follows a prescribed trajqory. This
problem was presented by Bayo et al. for robot
manipNator in frequency domain, by Das, Utku and
Wada for trusf, structures in time domain, and by
Aasada and Ma , for manipulators in time domain.
These fpethods are of iteratwe in nature. Bay0 and
Moulin
solved linear inverse dynamics problem
with Coriolis and centrifugal forces neglected. In
this paper we consider two types of flexible
inverse dynamic problems. In the first one, joint
torques are determined such that the tip trajectory
error is annihilated. In the second one joint
torques are determined such that the joint angles
in the flexible configuration are equal to those in
The first two assumptions can be
met for many
space manipulators by design. For example, the
space crane command torque profiles can be designed
such that amplitudes of flexible motion are
0
Copyright
American Institute of Aeronautics and
Astronautics, Inc., 1990. All rights reserved.
725
small"; also, as studied in Ref. 18 and Ref. 19,
for 300 ft crane, the rotation rate of each link is
less then 0.0024Hz while the link fundamental
frequency is 0.8Hz. When a link rotates, its
natural frequencies are shifted; when the rotation
rates are small, the changes in frequencies can be
ne lected. Hence, natural frequency shifts due to
lin rotation rates are ignored in this paper.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396
'k
Nonlinear effects are due to large manipulator
articulations and to the presence of Coriolis and
centrifugal
forces.
The
effects
of
large
articulations are removed from the derivation of
equations of motion with respect to the manipulator
rigid
body
configuration. Elastic deformations
measured
with
respect
to
the
rigid
body
configuration
are
linear;
hence,
linear
time-varying system is
obtained.
For
small
vibrations, Coriolis and centrifugal forces in
flexible configuration are very close to these of
rigid body configuration. In recent studies and
linF
flexible
numerical
simulations of
two
manipulators, Padilla and von Flotow presented
that for small rotation rates one can model a
system with reasonable accuracy by keeping the
';near
equations
together
with
Coriolis
and
centrifugal forces determined from the rigid body
motion.
The manipulator dynamics is described by the
following equations of motion
+~p~,t).
xe(o)=xe0, ;,co)
M(XJ;: = T ~
According to the second assumption, the
centrifugal and Coriolis forces in the flexible
body configuration are approximately the same as
those in the rigid body configuration
F,cXo5J F,cx,,i)
(4)
Introducing (3) and (4) to (1) one obtains
M(x);*
+C(x);
0.
+K(x)x=f(xr,xr,xr)
(5)
x(0)=xo, i(0)=vo,
where
In
f(xr,xr,xr)=Te
0 0
+FC(x,,;)-M(x);:.
q.(5),
M(xJ=M(x)
is used due to small elastic
deformations. The vector f is determined fully from
the rigid body configuration. It can be decomposed
into vectors of joint torques T and the nodal
forces and torques F
f=T+F
wbere T = P T 0IT, F=[O F:].'
equation (5) becomes
With the above notation
M(xl)i* +C ( X ) +K(xl)x
~
=T(xr,xr,xr)
m o +F(xr,xr,xr)
0
0
0.
(6)
x(O)=xa, Z(0)=vo,
=vo0, (1)
wbere xe is the generalized displacement vector, To
is a vector of external torques, M(xJ is a
nonlinear mass matrix, Fe is a vector of nodal
forces and torques caused by centrifu al and
Coriolis forces as well as elastic de ormation
forces. The generalized displacement vector is of
dimension n =nr+ne consisting of manipulator joint
angles ao, of dimension nr and elastic link
displacements q,, of dimension ne,
For a given tip trajecto7, the
inverse kinematics relates uniquely
rigid configuration to time. Matrices in
time dependent, and eq.(6) is a linear
differential equation
f
rigid body
manipulator
eq.(6) am
time-variant
(74
M(t);O+C(t);+K(t)x=T(t)+F(t)
x(0)=xo, i(0)=vo.
Finally, with the generalized displacement
vector divided into joint and nodal components
XT=[eT, qT]
Let xr be the rigid body displacement vector,
the elastic displacement vector with
and x=x;xr
respect to the rigid body configuration. For small
elastic deformations, the link deformation forces
are linear in x. Thus
eq. (7a) is re-written with the
dropped for clarity,
{eT(o)
where K, C are the stiffness and dampin matrices,
respectively, and Fc is the centrikgal and
Coriolis forces vector. The above flexible model,
unlike the rigid body model, has two independent
angles BIiand e,,
and
two independent torques,
Tfi and Tfi at each joint i, see Fig.lb. The angles
i
satisfy
the
relationship:
at
joint
a01.-aA
=e,-efi.
(2b)
T T T
qT(o))T={eoqo) ,
argument
t
is
{BTco) 4T(o))T=(~T,f>'
(7b)
For free rotating joints, the stiffness matrix
is positive semi-definite, Similar statement can be
applied to the damping matrix, that is, zero
friction results in positive semi-definite damping
matrix. The mass matrix, however, is always
positive definite since rotary inertia always has
non-zero value.
726
I
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3. Flexible manipulator inverse dynamics
A solution of the forward dynamics problem is
obtained from Eq.(7): flexible deflections of the
tip and links, as well as joint angles are
determined given the applied forces and torques.
For most applications, however, is desirable to
command the tip or other parts of a manipulator to
follow a prescribed trajectory. This forms the
inverse dynamics problem. In this problem, one
determines the required forces and torques such
that the tip or other parts of the system follow a
specified trajectory. The most obvious way to
define the inverse dynamic problem is to keep the
tip elastic displacements zero, and to determine
joint torques to meet this requirement. Our
investigations and simulations show that the
solution of this problem is unstable. Thus, another
way to solve the flexible body inverse d amics
problem is needed. The new approach is t at the
joint torques are determined such that the joint
angles of the flexible model equal the joint angles
of the rigid body model. This approach yields a
stable and robust results, and at the same time the
tip displacement error is relatively small. In the
following we will concentrate on the latter
approach.
f
Consider a manipulator with joint 1 connectin
link 1 to the base, and each of the remaining
joints connecting two separate links together. Let
e,, TI be the angle and torque at joint 1. In the
manipulator model, each joint that is
connected to the base is associated with
iodependent angles Ofi and e,,
and
independent torques, Tfi, T,
not
two
two
1 0
0
0 1
0
O 1
0
ne
where Ik is an identity matrix of dimensions k x k .
Note that PIF=F, thus the left multiplication of
eq. (7) by P1 gives
P~MY
+P~C;+P~KX
=T+F
The equality of joint angles in the rigid (ari
in Fig.lb) and flexible configuration (aei in
Fig.1b) requires that left and right angles are the
same, say 8, (8,=0 for i = l )
Bfi=8,=0~, and Od=0
(9a)
Manipulators joint torques are t ically internally
reacting with equal and opposite epects on the two
attachment points, therefore one independent torque
per joint is sufficient, say Tai, such that
With conditions (9),
vector is now
the generalized displacement
To 1
In this cas the displacement and force vectors in
eq47) are
where
Next, the transformation of the torque vector
is introduced, such that
and eq.(8)
.-
~.
fix +ex +gx
=T+F
where
fi=PIMP2,
where
e! =P1CP2,
=PIKPz,
?' =[TTO0IT(11b)
and TT=[TT TT]. In the above equation x, and T, arc
unknown variables. Defining the unknown vector p:'
x;lT, the forcing vector FT.=[O Fz], and
727
6. Compute T,(r> from q.(l2b).
4. Flexible manipulator control
q.(ll) can then be transformed into the following
The flexible manipulator is controlled by
joint torques T, and forces u, as shown in Fig.2.
Joint torques consist of feedforward torque T,
pre-computed from the inverse dynamics and feedback
torque Tb which is proportional to the error signal
a;ar, (deviation of flexible body joint angle from
the rigid body joint angle). Flexible displacements
of links are damped through the member-stiffening
loop, by sensing member bending angles and
actuating against the bending torques, in which
sensors and actuators are collocated.
form
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or, equivalently
and
T = = M ~ ~+cl
~ ; : 21;1 +K,
From eq.(12), the manipulator equation
motion can be rewritten in the state-space form
z ~ s
Z =AX + B ~ U+ B ~ F ~ ,
From eq.(l2a) one obtains the flexible body
displacements, velocities and accelerations of the
links; eq.(l2b) gives the required control torques.
of
y=cx
where u is the control force, and F, represents the
disturbance forces (Coriolis and centrifugal). With
a feedback loop, let
Note that for stable system described by
eq.(ll), the control torques of eq.(l2) exist and
their amplitudes are bounded.
u=Gy.
Solutions of the forward and inverse flexible
dynamics problems are computationally intensive,
since their differential equations are time-variant
(e.g. mass matrix is joint an le dependent). A
solution procedure is proposed %elow. Since both
forward and inverse dynamics problems are described
by a similar set of equations (cf. eqs.(7a) and
(12a)), only a procedure for solving the inverse
dynamic problem eq.(12) is described here.
One has to determine the gain matrix G such that
the tip trajectory satisfies specified performance
criteria.
he root locus methodm is used for
determining the closed-loop gain G. The gain is
computed for both the full system as well as thr
reduced order model. The ain for the reduced order
model is determined as kllows. For time interval
ti, let (A,B1,C) be the state space representation
of the system, and its balanced representation be
(Ab,Bb,Cb). Limited-time balancing method is used
for balancing the system in the finite time
interval ti(see Ref.21 and 22).
The manipulator model is a slowly time-varying
one, the link rotation rates are small, compared
with their fundamental frequencies. Thus, the
solution of eq.(l2a) can be obtained by assuming
piece wise constant values for the matrices in the
time-varying equations. Let At be a small time
interval, and let i=O,1, ...,m, such that mdt=r, r
is the tip fly-time.
Denote ti=idt, and time
interval ri=[ti, ti+J. For the time interval ri
one solves eq.(l2a) with constant matrices fi.(t>,
“t),
ti), with the initial condition x$t>
determined from previous interval, and x$O)=O.
Then from eq.(l2b) one finds Tx(7>.
For lightly damped flexible structures with
separate poles,
the
matrix
Ab is
almost
block-diagonal with 2 x 2 blocks, see Refs. 23,24,21
Ab=
llAKj
112
~ ~ A b i i ~ ~ z ~IIz9
~Abjj
..
The procedure for solving the inverse flexible
dynamics problem is summarized as follows:
1. For gwen tip trajectory compute the rigid body
joint
angles
(which
form
a
rigid
body
configuration) and torques by a standard rigid body
inverse kinematics and dynamics procedure.
2. Use the joint angles obtained in step 1 for time
ti to compute the mass, damping and stiffness
matrices.
3. Assemble matrices to obtain
(12).4. Com Ute
the rigid body Coriolis and cenmxgal forces b m
the rigid body configuration for time interval ri,
and apply them to the flexible model, q.(l2a).
5. Solve q.(l2a) for the time interval ri with the
initial condition x,(t>.
728
i j = 1,. ,n2, i # j,
n2 =n/2,
11. llz-spctral norm.
Matrix Bb consists of two-row blocks BE, and
matrix C, consists of two-column blocks CK
associated with blocks AM
wtl,..., BTb d
9
cb=ICbl .,cml
9.
The system pole shift due to output feedback
is studied by means of pole mobility factor, scc
Appendix. Partition the balanced state variable x,,
into two parts: x, and xt, thus x ~ = [ x ~ , x ~where
],
x, is a k x 1 vector, and xt is a (n-k)x 1 vector.
Partition Ab, Bb,Cb accordingly
- -
r
to the rigid body configuration.
with res*t
Applying the feedforward torques, the simulated
manipulator elastic tip elastic displacements .ire
sb;wn in Fig.4, with solid line for the x-Comtx\nmt
and dashed line for the z-component. The
feedforward torques are shown in Fig.5. One can
notice small oscillations during the transition.
This is due to the small incompatibility between
the models of two sequential time instants. The
smaller the time intervals is, the smaller
discrepancies will appear.
The subsystem (A,,B,,C,)
consists of components
with
high
mobility
poles,
and
the
subsystem(At,Bt,Ct) consists of components with low
. With collocated sensors and
actuators t e pole mobility is characterized by the
mobility
factor pi, defined by eq.(A7) in the Appendix
res
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p,=up,
Note that the feedforward torques act on the
joints,
lowering natural
frequencies of
the
manipulator, especially for the low fr uency
modes, see Fig.6. As a result, the crane w% the
feedforward torques is softer then the crane
without feedforward compensation.
(14)
With output feedback for (Ab,Bb,Cb)
The robustness of the feedforward compensation
algorithm to system parameter variations is checked
as follows. Let Ma, KO, Co be the nominal values of
the
mass,
stiffness
and
dampin
matrices,
respectively; let their actual values %e M=aMo,
K=aKo, C=aCo, where a is a scalar. The compensation
torques are determined for the nominal parameters,
while the manipulator dynamics is obtained for the
actual parameters. Let s,=[x,,
z,] be the tip
elastic displacement of a manipulator for actual
parameters, with the applied feedforward torques
determined for the nominal parameters. Let
sto=[xm, z,,] be the tip elastic displacement for
nominal parameters, then p= IIs,II Jlls Jl Qo is a
measure of robustness of feedforward compensation.
The feedforward control is considered robust if psi1
(tip displacement for the deviated parameters is
about the same as for the nominal parameters). The
robustness indicator p is determined for various
values of the multiplier a ran@ng from 0.5 to 2.
The plots of p ( a ) , corresponding to the deviation
of the mass, stiffness and damping matrices are
shown in Fig.7. The plots indicate that the
feedforward control algorithm is fairly robust to
stiffness
and
damping
variations
(with
0.998<p<1.001 for 100% variations from their
nominal values), and less robust to mass matrix
variations (with 0.5 < p < 2 for 100% matrix variations
from the nominal value, and 0.909 < p < 1.11 1 for 10%
variations from the nominal value).
u=k C x
cl
b b
diagonally dominant and BtCt small one obtains
the following closed loop matrix Ac
b‘
A =Ab+kqBbCbsdiag(Ar+kqBrCr,At).
This expression shows that eigenvalues of Ar
are shifted about the same amount as the related
eigenvalues of Ac, while eigenvalues of At remain
nearly unchanged. Therefore, the gain obtained. for
the reduced order model is nearly the same as that
of the full model. Note that the feedback loop not
necessarily moves poles of the most observable and
controllable components, or components with the
highest cost, but those with the highest mobility
factor, which is a ratio of square of the component
cost and the joint measurez of controllability and
observability, that
is
u,ly,.
This will be
illustrated in the next section.
5. Applications
In-plane dynamics of a two-link flexible
manipulator, which is considered as
a two-link
version of the space crane, Refs. 18, 19, is
considered. The two links are assumed identical,
see Fi .3a, with the following physical properties:
length f = l l 8 l . l in, cross section area A=54.72 in ,
bending moment of inegia I=14510 in4, modulus of
elasticity E=33.8 x 10 lb/in2, shear modulus G=13
x 10 lb/in*, linear mass density mo=0.009631 lb
sec2/in. The payload at the tip weighs 40000 lbs.
The finite element model of each link consists of 3
beam elements. The mani ulator’s initial
is defined by joint angles 4=n14 rad, and
rad. The manipulator performs lifting maneuver,
with tip vertical movement h=800 inches in 20 sec.
The simulation results presented here were
obtained for no external disturbances applied to
the model. Hence the error a;ar--0 and the member
stiffening feedback loop is the only loop that is
active. The gain of this feedback loop is
determined by the root-locus method for the full
and reduced order model. Assuming the output
feedback u=ky, where k is a scalar, the plot of the
root locus of the full system is shown in Fig.8a.
One can scc that middle-frequency poles are
significantly moved, while low and high frequency
poles remain almost unchanged.
The manipulator dynamics and inverse dynamics
were simulated for 40 sec time interval. This
interval was divided into 21 smaller intervals, the
first 20 of them are of length 1 sec, while the
last one of length 20 sec (no update is necessary
in the last interval, during which the rigid body
configuration of the crane is constant). The rigid
body configurations, for which the crane model is
updated are shown in Fig.3b. As mentioned earlier,
the manipulator clastic displacement is measured
In the reduced order root-locus technique the
system Hankel singular values and component costs
are determined and shown in Fig.9a,b. The plots
indicate that the low fr uency modes are the most
observable and controllab e, and have the highest
cost as well. The plot of Fig.8a indicates,
cost
and
high
however,
that
those high
7
729
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controllability and observability modes are h o s t
unchanged when output feedback is applied. Instead,
the middle frequency modes are significantly moved.
The pole mobility factor defined as a ratio of the
square cost to the Hankel singular value, eq.(14),
is shown in Fig.9c. The plot indicates that the
middle frequency components 5, 6, 7, 8, and 9 have
the highest pole mobility ratio. These components
have been chosen for the reduced order model. The
root-locus plot for the reduced 5 component (10
states) model is shown in Fig.8b. The root
trajectories are similar to those of the full order
model.
'R.P. Judd, a d D.R. Falkenbw? "Dynamics of
Nonrigid Articulated Robot Linkages , ZEEE Trans.
Autom. Control, vol.AC-30, 1985, pp.499-502.
7
G. Naganathan, and A.H. Sod: "Coupling Effects
of Kinematics and Flexibility in Manipulators",
Znt. J. Robotics Research, ~01.6, 1987, pp.75-84.
8
K.H. Low: "Solution Schemes for the System
Equations of Flexible Robots", J. Robotic Systems,
VO1.6, 1989, pp.383-405.
9Y. Huang, C.S.G. Lee: "Generalization of
Newton-Euler Formulation of Dynamic Equations to
Dynamic
System,
Nonrigid
Manipulators", J.
~01.110, 1988,
Measurement,
and
Control,
pp.308-3 15.
The open-loop response (yo), and closed-loop
response (y) are compared in Fig.10, for k=2*ld.
The performance, defined as p= ((yc((2/((yo((z
is good,
namely p < 0.01.
lop. Tomei, and A. Tornambe: "Approximate
Modeling of Robots Having Elastic Links", ZEEE
Trans. System,Man, Cybernetics, vol.CAS-18, 1988,
pp. 831-840.
6. Conclusions
I1
M.A. Serna, and E. Bayo: "A Simple and
Efficient Computational Approach for the Forward
Dynamics of Elastic Robots", J. Robotic System,
~01.6,1989, pp.363-382.
A novel linearized approach for solving
flexible manipulator dynamics
is proposed in the
paper. This approach has greatly reduced the
complexity of tbe control design and simulation
costs. Based on this approach, the inverse dynamics
problem is defined and solved. The forward
compensation torques are determined with the joint
angles in the flexible body configuration matching
the angles in the rigid body configuration. The
combined feedforward compensation and feedback
control is robust
to the
model
parameter
uncertainties
and
resulted
111
satisfactory manipulator performance.
12
E. Bayo, M.A. Serna, P.Papadopoulus, J. Stubbe:
"Inverse Dynamics and Kinematics of Multi-Link
Elastic Robots. An Iterative Frequency Domain
Approach", Report UCSB-ME-87-7, 1987.
I3S.K. Das, S. Utku, and B.K. Wada: "Inverse
Dynamics of Adaptive Structures Used as Space
Cranes", JPL Znternal Document 0-6489, 1989.
14H. Asada and 2.-D. Ma: "Inverse Dynamics of
Flexible Robots ", Proc. I989 American Control
Conference, Pittsburgh, 1989, pp.2352-2359.
ACKNOWLEDGEMENT
This
research was performed at the Jet
Laboratory,
California Institute
of
Tec ology, under contract with the National
Aeronautics and Space Administration.
I5
Pr-
E. Bayo, and H. Moulin: "An Efficient
Computation of the Inverse Dynamics of Flexible
Manipulators in the Time Domain", ZEEE Robotics ana'
Automation Conference, pp.7 10-715, 1989.
16
J.J Craig, Introduction to Robotics. Mechanics
and Control. Addison-Wesley, Reading, 1988.
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Structures, Wiley, New York 1990.
4
X. Cyril, J. Angels, and A.K. Misra:
"Flexible-Link Robotic Manipulator Dynamics", Proc,
I989 American Control Conference, Pittsburgh, 1989,
pp.2346-235 1.
Control
of
21
W. Gawronski, and J.-N. Juang: "Model Reduction
for Flexible Structures". In: Control and Dynamic
System, ed. C. Leondes, vo1.35, Academic Press,
New York 1990.
5
M. Benati, and A. MOKO: "Dynamics of Chain of
Flexible Links", J . Dynamic System, Measurement,
and Control, ~01.110, 1988, pp.410-415.
22
W. Gawronski, and J.-N. Juang, "Model Reduction
in Limited Time and Fr uency Intervals", Int. J .
System Science, v01.21, 1 9 8 , pp.349-376.
730
23
C.Z. Gregory, Jr., "Reduction of Large Flexible
Spacecraft Models Using Internal Balancing Theory",
J. Guidance, Control and Llynamics, vo1.7, 1984,
~p.725-732.
Hankd singular values y,,
see Ref.25,
characterize
joint
controllability
and
observability of system components, while coats,
ai, see Ref.26, characterize the participation of
the components in the system output. They may be
considered as reasonable measures of pole mobility,
therefore we relate them to the pole mobility
factor. For flexible structures, see Refs.21 or 24,
"W. Gawronski, and T. Williams: "Model Reduction
for Flexible Space Structures", J . Guidance,
Control and Dynamics, vo1.14, 1991.
2s
K. Glover: AU
Optimal
Hankel
Norm
Approxinbfbtions of Linear Multivariable systems and
Their L
Error Bound", Znt. JoumZ of Control,
~01.39, 1984, pp. 1115-1193.
where Ai=2Cioi is a half power frequency, hence,
from (A5) and (A6) one obtains
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396
%.E.Skelton, and A. Yousuff "Component Cost
Analysis of Large Scale Systems", Int. Journal of
Control, vo1.37, 1983, pp.285-304.
Pi =<IYi=yiAi =uidd =0.5YiAi =YiCioi
where Yi=IIH(o,)l(, is a spectral norm of the output
at frequency o=oi. The determination of Y, and A ,
for a single-input single-output system is shown in
Fig. 11.
APPENDIX.
Pole mobility for symmetric flexible structures
Consider a linear system represented by a
triplet (A,B,C), where A is n x n , B is n x p , C is
qxn, and n is an even integer. A linear system with
matrix A having separated complex poles with small
real parts exhibip characteristics of a flexible
structure. If C=B S, where S=diag(l,-1, ...,1,-1) is
a sign matrix, then the system (A,B,C) is said to
be a symmetric one. For example, any flexible
structure with collocated sensors and actuators is
a symmetric system. Furthermore, let (A,B,C) be a
balanced representation of a symmetric flexible
structure. In this case, matrices A and BC are
almost block-diagonal with 2 x 2 blocks, compare
Ref.21 and 24,
A 1diag(Ai),
.
BC Idiag(B,Ci), i = 1,. . $2
One can see clearly that the pole mobility
depends neither on system controllability and
observability properties, nor component cost. It
rather depends on their ratio, or combination of
Hankel singular value and half power frequency, M
combination of cost and half power frequency.
Although the above properties were derived for
symmetric flexible structures, they are also true
for broader class of flexible structures, as long
as the product BC, in the balanced representation,
is almost block diagonal. For
single-input single-output system satis es ta9y
hs
requirement, and many non-symmetric systems as
well.
exmte,
(Al)
where n2=n12,
and Bi is two-row block of B, Ci is two-column
is
block of C, oi is the i-th natural frequency,
i-th modal damping coefficient. In the balanced
ral norms of those blocks are
representation
q u a l to each ot er, lIBil12=IlCiIIz.
ci
SF
The output feedback loop with a scalar gain k
results in the closed-loop matrix Ac=A+kBC, or
A Idiag(A,), where
The almost diagonal property of A and BC allows one
to characterize the shift of the i-th pole, per
unit gain by the pole mobility factor pi
Using (A3), (A4) becomes
Pi
-
(A7)
llBiCiIl2
731
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1200
joint 2
Fig.1.
aentrrlizod
manipulator
coordinates
of
a
flexible
Fig.3. Two link space crane and its configurations
during slew maueuver
Fi.2. Plexibk manipulator control 0013CCpt
732
1.8
-
.,
8 1.6.....
c)
0
;a” 1.4-
mass
A
cl)
a
12-
stiffness
and damping
...
:
1
1-
0
a -20Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396
.r(
c)
7.8
-
0.6
-
..........
... ...
..... ......
.................
0.4
-30
time, stc
Fig.4. Tip elastic displacements
Fig.7. Robustness indicator
xi07
I,
full order model;
1500
loo0
.......
..... ..........
.
-1sOo
I
0
5
10
15
20
25
30
35
40
--2OOo
-1600
time, SeC
-1400
l
-1200
o
-lo00
-500
0
o -800
W
0
Re
Fig.5. Feedforward joint torques
1500
-
loo0
-
lo’
reduced
order m~Ie1.
without torques .............
...
-
....
-............
J
Fig.6. Crane natural frequencies
Fig.8. Root-loci for
models
733
full
and
reduced
order
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‘
-6
0
-30;
5
10
15
20
2.5
30
35
30
5
10
15
20
25
30
35
40
time,
x10’
25,
2
I
I
Sac
1
I
n
Fig.10. Tip elastic displacement for feedforward
control only (solid line) and for feedforward and
feedback control (dashed line)
component no
Fig.9. Costs, HanLel singular
mobility for tbe crane model
values,
and
2
pole
I
I
3
4
5
I
I I l l
6 7 8 9 1 0
20
0
Fig. 11. Determining Hankel singular values from a SISO transfer function plot.
7 34